Household Search and the Aggregate Labor Market

Transcription

1 Household Search and the Aggregate Labor Market Jochen Mankart and Rigas Oikonomou Department of Economics London School of Economics and Political Science This draft: October 2010 Abstract Sharing risks is one of the essential economic roles of families. The importance of this role increases in the amount of uncertainty that agents face and the degree of financial market incompleteness. We develop a theory of joint household search in frictional labor markets under incomplete financial markets. Couples households can insure themselves by savings and by timing their labor market participation. We show that this theory can match one aspect of the US data that conventional search models cannot match; that whilst aggregate employment is pro-cyclical and unemployment counter-cyclical their sum, the labor force is acyclical. In our model, and in the US data, when a family member loses her job in a recession the other family member joins the labor force to provide insurance. We also explore other important implications of our theory for the aggregate labor market. Our analysis offers new insights for the cyclical behavior of the labor wedge in models of heterogeneous agents and wealth accumulation. JEL Classification: E24, E25, E32, J10, J64 Keywords: Heterogeneous Agents; Family Self Insurance; Labor Market Search; Aggregate Fluctuations We are indebted to Albert Marcet, Chris Pissarides and especially Rachel Ngai for their continuous support and guidance. We also benefited a lot from the comments of Arpad Abraham, Francesco Caselli, Antonia Diaz, Matthias Doepke, Wouter Den Haan, Nobu Kiyotaki, Omar Licandro, Alex Michaelides, Salvador Ortigueira, Monika Piazzesi, Franck Portier, Martin Schneider and Michelle Tertilt. We are also grateful to seminar participants at the LSE, St. Gallen, the XV Workshop on Dynamic Macroeconomics in Vigo, the European Workshop in Macroeconomics in Munich, the Econometric Society World Congress in Shanghai and the 25th Conference of the European Economic Association in Glasgow for valuable comments. address: j.mankart@lse.ac.uk address: r.oikonomou@lse.ac.uk 1

2 1 Introduction In March 2009 there were roughly 13 million unemployed workers in the US economy. A large fraction (40%) of these workers were married. Married men accounted for 25% of total unemployment and married women for 15%. 68% of all US workers participated in the labor force; for married women the analogous participation rate was 61%. Economic decisions such as whether or not to work and whether or not to search for job opportunities in the labor market, are surely made jointly in the family. Moreover when financial markets are incomplete, as they are in the real world, these decisions are influenced by the incentive of households to insure against shocks to their labor income. Unemployment is such a shock and families are an important insurance device against it. With very few exceptions these features have not been part of the literature of search models of the labor market; nor have they been part of the literature of models of heterogeneous agents and wealth accumulation. Take for instance the baseline model of Krusell & Smith (1998) in the latter literature. In this economy workers face idiosyncratic labor income risks, such as the risk of unemployment, and they can self-insure against them by trading claims on the aggregate capital stock. Much less common is the idea that a considerable amount of employment risk diversification can be provided within the family. Moreover in Mortensen & Pissarides (1994) and in the considerable literature of search and matching models of the labor market, households are also viewed single agents decision units. We present a theoretical framework where financial markets are incomplete, labor markets are subject to search frictions and households are formed by two members that make labor supply, search and savings decisions jointly. We use this framework to investigate the effects of family self insurance on the aggregate labor market outcomes. In particular we show that our model can match one aspect of the US data that conventional search models cannot match; that whilst aggregate employment is very procyclical and unemployment countercyclical, their sum, the labor force is nearly acyclical. Search models that allow for endogenous participation in the labor force have a hard time in matching these facts. As we explain they typically produce a very procyclical labor force and we argue that what they are missing out on is to assign an important role to family self insurance. To understand this last point consider the following realistic example: Assume that a couple has one of its members employed and the other member is out of the labor force. This is a pattern of intra-household specialization that we observe very frequently in the data. Usually primary earners in US households are husbands and secondary earners wives. Assume also that the economy is in a recession, when the separation rate is higher and the job finding rate is lower. If the husband loses his job in a recession, household income suffers a big shock. Moverover if financial markets are incomplete, income losses have an impact on 2

3 consumption; but joint search can provide an important buffer against these risks. The wife can join the labor force, and search actively in the market, to maximize the chances that the household will have at least one of its members employed next period. In section 2 of our paper we show that similar adjustments of labor supplies at the household level are a feature of the US data. We show that some household members time their flows in and out of the labor market to provide insurance, and once we remove this insurance effect their labor force participation becomes considerably more procyclical. We also draw on several studies that document the magnitude of the added worker effect, the behavioral response of female labor supply to spousal unemployment (for example Lundberg (1985) and Stephens (2002)). Then we turn to the theory. We construct three general equilibrium models with search frictions in the labor market and shocks in individual (idiosyncratic) labor productivity. There are incomplete financial markets as in Krusell & Smith (1998). We first present a model where households are bachelors. This is a natural starting point that helps us to examine transparently the role of family insurance and to describe the economic environment. Subsequently we present two models where families are couples. In the first one we assume that the two agents that form the family are ex ante identical, meaning that they have the same potential labor income and confront the same frictions in the labor market. In the second we populate each household with a husband and wife, and we calibrate labor market risks and search frictions for males and females separately; we also introduce differences in the gender price of labor, a gender pay gap. We consider these two different structures because the notion that families pool resources and adjust labor supplies is central in our theory. When family members have the same potential labor income there is possibly too much insurance relative to the data. When we model households as couples with a husband and a wife we address this concern. We compare the aggregate labor market allocations in these environments when we introduce business cycle fluctuations. If family self insurance is important in the models as it is in the data, then bachelor households should produce a more procyclical labor force. This is what we find; the three economies share similar cyclical properties for aggregate employment unemployment, but they deliver very different predictions for the cyclical behavior of the labor force. In both couples economies the contemporaneous correlation with GDP is nearly 50% smaller than that of our bachelors households model. The couples economies match the cyclical properties of this statistic as in the US data. We show that our models are consistent with the data in a number of dimensions. First, all of our economies can match very well the patterns of worker reallocation (worker flows between employment, unemployment and inactivity), and they can account for the idiosyncratic labor market risks that agents face throughout their working lives. Second our couples economies can account for the patterns of household specialization in market work and leisure and the 3

4 distribution of household members across labor market states. A large part of the success of our model in matching the US business cycle in the aggregate labor market, is due the assumption that risk sharing in the macro-economy is imperfect. This is the very important difference between our approach and previous work in the literature that studies the cyclicality of labor force participation in search models. For instance Tripier (2004) uses the search and matching model as in Merz (1995) and Veracierto (2008) considers a version of the equilibrium unemployment theory of Lucas & Prescott (1974) with undirected search and endogenous separations. Haefke & Reiter (2009) augment the Mortensen & Pissarides (1994) model with home production, but they assume risk neutrality. All of these models generate a very procyclical labor force, because the notion that families adjust their labor supply to provide insurance is absent. Instead we interpret movements in labor market flows over the business cycle partly as a behavioral response of household members to changes in unemployment risks. Finally, we explore another important implication of our theory. There is a growing literature that studies the cyclical properties of the wedge derived from the optimality condition that links consumption and hours in real business cycle theory. One strand of this literature views labor market wedges as a by product of aggregation of individual policy rules, in models of heterogeneous agents and wealth accumulation (Chang & Kim (2007)), and another strand as a symptom of the failure of labor markets to clear when there are frictions (see for instance Hall (2009)). Our theory combines both of these features. We find that although incomplete financial markets models can generate cyclical wedges when agents are bachelors they cannot do so when families are larger. Even with as few as two agents in the household the labor wedge has cyclical properties very different from the US data. Instead we argue that labor market frictions are more crucial. Only very recently a handful of papers highlight the role of family labor supply as an insurance margin against idiosyncratic labor income risks. Chang & Kim (2006) develop a framework where households consist of two members (a male and female) and use it to understand how individual labor supply rules affect the value of the aggregate elasticity of labor supply. Attanasio, Low, & Sanchez-Marcos (2005) quantify the welfare benefits from female labor force participation when income uncertainty increases in a model with incomplete asset markets. Attanasio, Low, & Sanchez-Marcos (2008) and Heathcote, Storesletten, & Violante (2008) analyze the effects of changes in the economic environment (such as changes in gender wage premia or changes in idiosyncratic labor income risks) on the historical trends of female labor supply. The difference from our work is that we emphasize the role of families in circumventing frictions in the labor market, whilst this literature overlooks the importance of frictions. This has the following implication: insurance against any income risk other than unemployment becomes less meaningful since families cannot readily assign the most productive agents to employment. The literature on the added worker effect that we summarize is corroborative to this interpretation of the risk 4

5 sharing role of families. Finally, Guler, Guvenen, & Violante (2008) explore the implications of joint search on optimal reservation wage policies. They use a stylized search model, while we build a general equilibrium framework with realistic heterogeneity that accounts for the observed labor market flows as well as the effects of shocks in aggregate productivity. The rest of this paper is organized as follows: Section 2 uses the estimated flows from the CPS to provide evidence that joint insurance and labor supply are key factors that explain the low procyclicality of the US LF participation. In section 3, we develop the bachelor household model and the couple household models. In section 4, we show and discuss the basic results and implications of our theory. Section 5 concludes and the computational details are relegated to the appendix. 2 The US Labor Market Table (1) summarizes the US labor market business cycle statistics. The data are constructed from the CPS and they correspond to observations spanning the years 1976 to They are logged and HP filtered and all quantities refer to quarterly aggregates and are expressed relative to a detrended measure of GDP. Unemployment is very counter-cyclical and more than 7 times as volatile as aggregate output. Aggregate employment has two thirds of the volatility of output at business cycle frequencies and is very procyclical. The labor force is not volatile and its contemporaneous correlation with GDP is low (.45). Table 1: US Business Cycle: Labor Market Statistics E U LF LF Couples LF Wives Aged 16 and Above Aged 25 to 55 σ x σ y ρ x,y Notes: The data are from the CPS and based on the years 1976 to They are logged and HP filtered and all quantities refer to quarterly aggregates and are expressed relative to a detrended measure of GDP. σx σ y is the volatility of variable x relative to the volatility of GDP. ρ x,y is the correlation of variable x with GDP. E is the employment to population ratio, U refers to the unemployment rate (number of unemployed agents over the labor force), and LF is the labor force (number of workers who are either employed or unemployed) over total population. Population is the total number of individuals in the relevant demographic group. The last columns of Table (1) present a breakdown of the relevant quantities into demographic groups that are of particular interest to us. For married couples aged 25 to 55 in our sample, aggregate statistics are no different than those of the full population (aged 16 and above). The labor force participation for this 5

6 demographic is even more acyclical (and hence even more puzzling from the point of view of theory) owning to the strong acyclical attachment of males in the sample, but also to the low contemporaneous correlation with GDP of female labor force participation. The volatility of both males (not shown) and females are higher than the aggregate volatility for this demographic group (column 4), because the labor force participation rates of wives and husbands in our sample are negatively correlated. We note that married couples in the data are an imperfect measure of our notion of couples in the model. Ideally we would like to have duads of agents that are linked with near perfect insurance opportunities within the family, and little insurance between, but the data preclude us from doing so. In the data families are extended beyond the household unit; singles are not really singles and frequently households consist of more than two agents who make labor supply decisions jointly. Despite these caveats, in what follows we will treat the joint search behavior of married couples as an ideal ground to provide evidence for our theory. 2.1 Implications for models: Fixed participation? The bulk of the literature of search theoretic models of the labor market assumes that agents can be either employed or unemployed at any point in time (see for example Mortensen & Pissarides (1994) and the considerable literature of search and matching models). We view this as a major shortcoming of the theory. In this section we show that flows between activity and inactivity are an important aspect of worker reallocation. We argue that the US data suggest that agents flow readily across labor market states and economic models need to embrace this feature of the data. The moments that we summarize here are also key targets for our models. In Table (2) we summarize the flows estimated from the CPS. Our sample includes all individuals, married or single, aged 16 and above. This is the population used by the BLS to construct aggregate labor market statistics for the US economy. These flows are the transition probabilities that an agent who is state i in period t will be in state j in period t + 1 where {i, j} {E, U, I}. E is employment, U is unemployment and I is inactivity (out of labor force). In the top panel (A) of the table we report the average transition probabilities for the population in the years In panels (B) and (C) we disaggregate our data into populations of males and females. Each month roughly 7 % of OLF (out of labor force) workers join the labor force (5% directly to employment), and 3 % of employed workers and 22 % of unemployed workers become inactive. Across the two genders, labor market flows seem strikingly similar, although males appear to have somewhat of a stronger labor force attachment than females. The employment population rate for males is 70% in our sample and roughly 50% for females. Aggregate employment population ratio is 60%. 6

7 Table 2: Flow rates Bachelors and Couples Panel A: All Agents Age 16 E U I E U I Panel B: Males Panel C: Females Age 16 Age 16 E U I E U I E U I Notes: The data are drawn from the CPS. These flow rates are averages over the period They are the monthly transitions from one labor market state in month t (rows) to another labor market state in month t + 1 (columns). E denotes employment, U unemployment and I inactivity (out of labor force). Table 3: Flow rates Married Couples Males Age Females Age E U I E U I E U I

8 In table 3 we look closer at labor market flows of married couples (males and females) aged between 25 and 55. We do so to dispel the suspicion that the results are driven by demographics; by focusing on this age bracket, we can partially control for schooling and retirement decisions. If anything the estimates in table 3 reinforce our conviction that individuals make frequent transitions between labor market states from one month to the other. The flows from inactivity to either employment or unemployment are even larger in this case (roughly 10% for males and 9% for females vs 7.8 % and 6.5% in table 2). Also 1% of males and 3% of females quit employment each month and exit the labor force. 1 These numbers are huge. Over our sample period there are more workers flowing from employment to out of the labor force than to unemployment; moreover if 35% of the US population are inactive and roughly 4% unemployed, there are more workers moving from out of the labor force to employment each month (1.7%) than from unemployment (1%). One of the important contributions of this paper is that it offers a framework that can be used to think about the determinants of these transitions. In the next paragraph we show that a large part of these flows is driven by the incentive of families to insure against unemployment. 2.2 How can we use the data to demonstrate our point? We show in this paragraph that family self insurance is important in the US data. We do two things. The first is that we estimate several limited dependent variable (linear probability) models to gauge the effect of the husband s employment status, on the wife s labor force transition probabilities; this allows us to control for some relevant aspects of heterogeneity. We also provide a summary of numerous attempts in the literature to determine the magnitude of the so called added worker effect, the behavioral response of female labor supply to spousal income 1 There is a large literature that documents similar facts using different micro data sets for the US economy (for instance Blanchard, Diamond, Hall, & Murphy (1990), Nagypal (2005), Davis & Haltiwanger (2006) and Shimer (2007)). We summarize a few relevant findings here. First Nagypal (2005) argues that around 40% of the transitions from employment to out of labor force are followed by a transition to employment in the next month. Some of these workers search for new opportunities on the job, and they obtain a new job but the starting date is postponed by a month. Second, flows from I to E are surely affected by time aggregation, since within a month, which is the interview horizon in the CPS, workers can move from inactivity to employment without having a recorded unemployment spell (e.g. Shimer (2007)). Finally Jones & Riddell (1999) argue that the behavior of passive searchers and marginally attached workers is important; for these groups they demonstrate that they have transition probabilities to employment that are nearly half as large as those of unemployed workers, and hence part of the flows between states U and I can be broadly interpreted as time variation in search intensity for these groups. These implications have already been explored in the literature; For instance it appears that adjusting the transition probabilities to embrace the idea that marginally attached workers should be treated as unemployed rather than inactive does not make a big difference in the estimated transition matrices, see Krusell, Mukoyama, Rogerson, & Sahin (2009a). In the models we try to match the flow rates as we document them in table 2. What is important is that our economies leave room for individuals to make frequent transitions between employment, unemployment and inactivity, so that the labor force is not fixed. 8

9 shocks. Both our own estimates and the related literature seem to be corroborative to our view that family labor supply is an important insurance mechanism against unemployment risks. Our second piece of evidence is that we use the data from the CPS to provide an answer to the following question: assuming that the employment status of married men does not fluctuate over the business cycle, how would female labor force participation behave? We perform a counterfactual experiment that can partially quantify the contribution of the joint labor supply on the cyclicality of the US labor force The Literature on the Added Worker Effect. We give a brief summary of a related literature that uses panel data to investigate the effect of unemployment spells experienced by the husband on the spousal supply of labor. Our reading suggests that at least with respect to data and methodology there are two strands in this literature. 2 First, there are models that use variation in annual hours of work to identify how the husband s recorded unemployment spells affect the wife s labor supply. There does not appear to be a consensus in this empirical work for the magnitude of the AWE. For instance Heckman & MaCurdy (1980) find a small but significant added worker effect, but Pencavel (1982) doesn t. The reason for this is twofold. First there are other sources of insurance (besides joint labor supply) that minimize the loss of income due to an unemployment spell. Cullen & Gruber (2000) show that unemployment benefits have a massive crowding out effect on family self insurance. Second, more recently Stephens (2002) argues that the empirical literature fails to identify unemployment spells that result in substantial earnings losses. He shows, using data from the PSID, that in families of displaced workers there are significant added worker effects. More related to our story is the subset of studies that use short run transitions across labor market states (employment, unemployment and inactivity). These studies tend to find significant added worker effects. Lundberg (1985) uses monthly employment histories from a sample of the Seattle and Denver Income Maintenance Experiments (SIME DIME) to conclude that if a husband is unemployed then the probability that his wife enters the LF increases by 25 %, and the probability of her leaving the LF is 33 % lower, and she is also 28 % less likely to leave employment for unemployment. 3 Further more Spletzer (1997) uses a sample from the CPS and estimates limited dependent variable models of 2 There is also a recent set of studies that focus on the responses of spousal labor supply to health shocks, such as Gallipoli & Turner (2008, 2009) for Canada and Coile (2004) for the US. This work documents the complete lack of an added worker effect, although, in the context of health shocks this has an obvious interpretation; since disability shocks entail an intra-household transfer of time, that allows wives to care for the their ill spouses, wives are unable to increase hours in the market to make up for the lost income. 3 This pattern seem to hold mainly for white families in Lundberg s dataset. For hispanics and blacks the added worker effect is not significant or in some cases the flows to unemployment 9

10 the probability that wives join the labor force on demographics and the husbands employment transitions. His estimates also show that wives increase their labor force participation in response to spousal unemployment. In table 4 we provide our own estimates; We use data from the CPS over the period , a longer and more recent sample than the one used by Spletzer (1997). Our sample consists of married couples aged between 25 and 55; the husband is employed at the beginning of the month and the wife is out of the labor force. We estimate the probability that the wife joins the labor force as a function of various demographic characteristics, month effects and the transition of the husband between employment and unemployment, if any. The results derive from a simple linear probability model; 4 The first column shows that conditional on the husband becoming unemployed, his wife is nearly 6% more likely to join the labor force in any given period. In column 2 we distinguish between the husband s reason for job separation. We have data on the respondent s past employment status and whether they lost or quit their previous job. In families where the husband looses his job, wives are 5% more likely to enter the labor force, than in those families where the husband quit, and 8.3% more likely than in families where the husband remains employed. This result is, off course, very much consistent with models of joint labor supply; an unanticipated fall in income is much more likely to induce a powerful added worker effect. It is also consistent with the findings of Stephens (2002). 5 We conclude this paragraph by noting two things: First when the relevant literature has asked the right question the answer has been conducive to our hypothesis and we verify that with our own empirical work. Second, although estimates such as those of Table 4 are useful because they control for observed heterogeneity they cannot quantify the impact of family search insurance in the cyclicality of the US labor force. This is something that we address in the next paragraph A counterfactual experiment. We use our sample from the CPS to provide an answer to the following question: Assuming that the employment status of married men does not fluctuate over the business cycle how would the labor force participation of their wives behave? Recessions are times when husbands are more likely to become unemployed and if so they are less likely to find new job opportunities. If female labor supply acts as a family insurance device, more wives will flow into the labor force during a recession than otherwise expected, making female labor force participation less pro-cyclical. We test the converse: if it were not for the cyclicality in for husbands and out of the labor force for wives are synchronized (mainly for black families). That aside Lunberg s conclusion is that there is still a significant added worker effect. 4 Probit regressions give similar results. See Appendix for details. 5 We get similar effects for other relevant transitions. Married women are less likely to exit from the labor force if their husband looses his job. They are also less likely to exit when their husband fails to find a job if at the beginning of period t he was unemployed. 10

11 Table 4: Added Worker Effect Variable LPM Model 1 LPM model 2 Husband EU (.00245) (.00349) Husband Job Looser.0504 (.0048) Age f (.0001) (.0001) Age 2 f/100 1e 5 1e 5 (2.18e-6) (2.18e-6) Age m (5.6e-5) (5.6e-6) Educ f (.00027) (.00027) Educ m (.00023) (.00023) White (.00012) (.00012) Black (.0015) (.0015) No of Kids (.00023) (.00023) No of Kids ( ) ( ) Time Dummies YES YES R Notes: The data are from the CPS for the years They are observations for families where at the beginning of month t the wife is out of labor force and the husband is employed. The dependent variable takes the value one of the female spouse becomes part of the labor force (employed or unemployed) in month t + 1. ***Indicates significant at 1 percent level. 11

12 husband s unemployment incidence, female labor force participation would be more pro-cyclical. For each period t we estimate the transition probability of a wife from state i to state j conditional on her spouse making a transition from state k to l (p f t (i, j k, l)), and the unconditional probability for the husband (p m t (k, l)). Due to data limitations we cannot define these probabilities for all relevant labor market states. For this reason we restrict our attention to i, j {lf, olf}, and k, l {e, n} meaning that wives can either be in or out of the labor force and husbands either employed or nonemployed. 6 We let n t (i, k) be the share of the population of couples with a wife in state i and a husband in state k. With these estimates we construct markov transition matrices over the relevant state space {lf, olf} {e, n}. We multiply the initial shares n t (i, k) with these matrices and we construct distributions of agents over one and three month ahead horizons. We consider two objects; The first, that we label n A t+s(i, j), is created by the product of n t (i, k) and a matrix whose typical entry is p f t (i, j, k, l)p m t (k, l). The second which we label n C t+s(i, j) is the result of the product of n t (i, k) with a matrix where we time average the husbands transition probabilities between employment and non-employment. 7 The labor force participation of married women is given by j n C t (lf, j) and j n A t (lf, j) under the two measures. We note that if family insurance is important in the data then j n C t (lf, j) should be much more procyclical than j n A t (lf, j). The reason is that under measure j n C t (lf, j) husbands are not more likely to loose their jobs in a recession; and they are not less likely to find new jobs if they become unemployed, in a recession than in a boom. If female labor supply is an important insurance device against unemployment (as the estimates of table 4 suggest) then by time averaging the husband s transitions, we reduce the insurance effect in a recession for US households. Table 5 summarizes the results from this experiment. We compare the relative standard deviations and contemporaneous correlation with a detrended measure of GDP. The first column refers to the cyclical properties of the labor force participation rate of married wives based on the actual population measure n t. Columns 2 to 3 and 4 to 5 compare the analogous objects based on the measures n A t and n C t, for one and three months horizons respectively. The result is both qualitatively and quantitatively encouraging. The cyclical correlation of labor force participation for wives jumps from.2988 to.3703 in columns 2 and 3 and from.257 to.3216 in columns 4 and 5 which roughly corresponds to a 25% increase in cyclicality. We do not interpret this 25% as a target for our models to match; the calculation of this section misses out on several important aspects of family self insurance, for example, that households 6 We focus on families where both spouses are between the ages 25 and 55. For this demographic group married agents account for roughly 60 % of the population. 7 We need both of these objects because small errors that compile over time make the comparison between n C t (i, j) and n A t (i, j) much more meaningful than between n C t (i, j) and n t (i, j). 12

13 Table 5: Experiments n t n A t n C t n A t n C t One Month Horizon Three Month Horizon σ x σ y ρ x,y Notes: The table shows the cyclical component of female labor force participation rates according to two measures: n C t corresponds to a markov transition matrix where the husbands transition probabilities (between employment and non-employment are kept constant; n A t allows for changes in the numbers of non-employed male spouses over time. might respond preemptively to increases in unemployment risks in recessions or with a lag, or the distinction between quits and separations might be important. Rather we give the following interpretation to our results: if the US economy was populated by bachelor households, then the labor force would be substantially more procyclical. We investigate whether our theory is consistent with this implication in section 4. 3 The model We develop three related models in which households face uninsurable idiosyncratic labor income risk. In section 3.1 a household consists of one agent, the bachelor. In section 3.2 families are formed by two ex ante identical agents. In section 3.3 we add further heterogeneity and we assign a gender (male or female) to each member of the household. 3.1 Bachelor economy We consider first an economy populated by a unit mass of strictly risk averse bachelor households; these agents are identical in preferences and they get utility from consumption c and disutility from hours working or from looking for job opportunities. We denote the discount factor by β and the period utility from consumption by u(c). At any point in time an individual can be either employed, unemployed or not part of the labor force. We assume that labor supply decisions are formed at the extensive margin and are subject to the frictions that impede instantaneous transitions across these labor market states. In particular employed agents are matched with firms in production and spend a fraction h of their unitary time endowment each period in market activities associated with a utility cost Φ(h). Every match operates a technology with constant returns 13

14 to scale so that without loss of generality we represent the total production in the economy as Y t = K α t (L t λ t ) 1 α. K t and L t denote the aggregate capital stock and labor input (per efficiency units); λ t denotes the level of TFP. We assume that λ t evolves according to the transition cumulative distribution function π λ λ = P rob(λ t+1 < λ λ t = λ). For non employed agents we assume that job availability is limited: We endow them with a technology that transforms units of search effort s t into arrival rates of job opportunities p(s t, λ t ) at a cost k(s t ) per unit of time. As we elaborate below on the basis of these optimal choices, we classify household members as either unemployed (active searchers) or out of labor force workers. We assume that individuals face idiosyncratic labor productivity risks and we summarize this in two independent stochastic processes; The first one, which we denote by ɛ, is an agent specific process, an own labor productivity component, that is a persistent state variable in the value function independent of her labor market status. We assume that ɛ evolves over time according to the transition cumulative distribution function π ɛ,ɛ = P r(ɛ t+1 < ɛ, ɛ t = ɛ). The second stochastic process is match quality; we assume that shocks arrive to this match quality at rate χ(λ t ) each period. When the shock hits, match productivity is driven to zero, which effectively leads the worker and the firm to separate. Financial markets are incomplete and agents can self insure by trading non contingent claims on the aggregate capital stock, earning a return R t each period, subject to an ad hoc borrowing limit a t a t. Wages per efficiency units of labor w t as well as rental rates R t are determined in competitive markets where the representative firm aggregates all inputs into the multipurpose final good. Aggregate capital K t depreciates at rate δ each period. Finally, we let Γ t denote the density function of agents over the relevant state space (of employment status, productivity and wealth). The law of motion for the distribution of workers is defined as: Γ t+1 = T (Γ t, λ t ) where T is the relevant transition operator The timing of events Each period t, and after the resolution of all relevant uncertainty, a non-employed agent chooses optimally the number of search units s t to exert. Her choice of s t maps into a probability p(s t, λ t ) of receiving a job offer in the next period. When this opportunity arrives the new values of the idiosyncratic productivity ɛ t+1 is sampled and the aggregate state vector {Γ t+1, λ t+1 } is revealed and the agent will decide whether she wants to give up search and become employed. Notice that given that all jobs entail a cost Φ(h) per period, the realization of the relevant state vector might not be such that the prospective match generates a positive surplus for the worker. In that case the agent continues to search in the labor market. Employed agents run the risk of loosing their jobs from two sources. First a fraction χ(λ t ) of all exiting matches terminate each period due to the arrival of the match quality shock. Second, the sampling of the new value of ɛ t+1 generates 14

15 the risk of separation. If ɛ t+1 is too low the worker may decide that it is not worthwhile to spend h of her time working and would rather be not employed next period. We assume that match quality (separation) shocks occur independently of the realization of ɛ Value functions Denote by V n and V e the value functions of a non-employed and an employed agent respectively. Also define Q e = max{v n, V e } as the outer envelope over the relevant menu of choices for the non-employed agent conditional on her receiving a job offer next period. Lifetime utility V n solves the following functional equation: V n (a, ɛ, Γ, λ) = max u(c) k(s) + β [ p(s, λ)q a a,s ɛ e (a, ɛ, Γ, λ )),λ subject to the constraint set 8 : + (1 p(s, λ)) V n (a, ɛ, Γ, λ ) ] dπ ɛ ɛdπ λ λ (3.1) a = R λ,γ a c. (3.2) Moreover the lifetime utility of an employed worker is the solution to the following functional equation:. V e (a, ɛ, Γ, λ) = max u(c) Φ(h) + β [ (1 χ(λ))q a a ɛ e (a, ɛ, Γ, λ )),λ subject to + χ(λ)v n (a, ɛ, Γ, λ ) ] dπ ɛ ɛdπ λ λ (3.3) a = R λ,γ a + w λ,γ hɛx c (3.4) A few comments are in order: First note that although the value function V n in equation 3.1 makes no explicit reference to unemployed or OLF workers, it summarizes both of these labor market states. Our classification criterion is the following: { < if s smin Worker is OLF s min Worker is Unemployed In words we classify an agent as unemployed if she chooses effort above a given threshold s min, and as out of the labor force otherwise. This mapping is 8 Notice that the distribution Γ becomes a state variable in the worker s value function. In order to forecast prices in the current context and to make optimal savings and labor market search decisions knowledge of Γ is necessary since this object determines the economy s aggregate capital stock and effective labor in the next period. 15

16 consistent with the notion that inactive agents search less intensively in the labor market. 9 Furthermore, we normalize the value of income for both unemployed and OLF workers to zero so that their consumption is financed exclusively out of the stock of savings. This assumption is made mainly to avoid the complications of having to talk about eligibility in government insurance schemes as it is not clear how benefits would be distributed across the population. For instance inactive workers in principle should not receive any replacement income but in our model there is a considerable amount of mobility between the two non employment states. Keeping track of benefit histories would add to the computational burden of our exercise. 10 In the Appendix we present the definition of the competitive equilibrium in this economy (see section A.2.1 for details). 3.2 Couples economy: Ex ante Identical Agents We now study the program of a couple in the same economic environment as described above. In this section a couple consists of two ex ante identical agents who pool their income and make labor supply and search decisions jointly. We adopt the unitary framework for simplicity and we assume that consumption is a public good in the household. We denote the time preference parameter for the household by β Value functions. We adopt the convention that the array (k, l) k, l {E, N} denotes a household whose first and second member are in states k and l respectively. Also it will prove useful to define the following objects beforehand: 9 The CPS classifies non employed workers on the basis of the following algorithm: First a non-employed respondent is asked whether he would like to have a job. Those who reply no are automatically considered as OLF workers. Those who reply yes are then asked to indicate what steps they have taken towards finding employment in the previous month. In particular they are asked to outline their methods of search; there are twelve such methods; for example workers can send out applications, answer job adds, enrol with a government employment agency. Those that have not used either of these methods but also those who search passively, for instance by reading newspaper adds, are classified as OLF workers. See Shimer (2004) for further details. 10 Arguably the unemployment insurance in the current context would crowd out family self-insurance, see Cullen & Gruber (2000), but it would also crowd out the precautionary role of assets, see Engen & Gruber (2001). Further more, although empirically one effect may not make up for the other it seems to be the case for the incomplete market model that we use here. For instance Young (2004) finds that the optimal level of benefits in an economy with frictions is always zero. The reason is that in general equilibrium, wealth accumulation minimizes the utility costs from the lack of government insurance. In the context of our model it seems likely that introducing benefits would only shift the regions in the state space where all the action takes place without any significant impact on the main conclusions. 16

17 Q en = max{v nn, V en } (3.5) Q ne = max{v nn, V ne } (3.6) Q ee = max{q en, Q ne, V ee } (3.7) These objects are the upper envelopes of the value functions and define the relevant menu of choices for our households. A household with one employed member can in any given period decide to withdraw her from the labor market and allocate both agents to search. This option is described in equation (3.5). Analogously, in (3.7) a household in which both members are employed, can withdraw both of them to non-employment, or keep one working or both. Each agent has her own idiosyncratic productivity and consequently household members differ in their productive endowments. But to conserve on the notation we denote by ɛ the vector of productivities of the members of a household. With these definitions we can represent the dynamic program of a household with two non-employed members as: + β V nn (a, ɛ, λ, Γ) = ɛ,λ max a a,s 1,s 2 u(c t ) i [ p(s1, λ)p(s 2, λ)q ee (a, ɛ, λ, Γ ) + p(s 1, λ)(1 p(s 2, λ))q en (a, ɛ, λ, Γ ) + p(s 2, λ)(1 p(s 1, λ))q ne (a, ɛ, λ, Γ ) k(s i ) + (1 p(s 2, λ))(1 p(s 1, λ))v nn (a, ɛ, λ, Γ ) ] dπ ɛ ɛdπ λ λ) (3.8) subject to: a = R λ,γ a c. (3.9) Optimal choices consist of current consumption and a pair of search intensity levels. Note that nothing precludes household members from setting s i s j although with standard convexity arguments this can only be the case if the productivity endowments ɛ i and ɛ j are unequal. Given s 1 and s 2 the household can anticipate that both of its members will receive a job offer next period with probability p(s 1, λ)p(s 2, λ) (in which case the envelope Q ee applies) and that with probability 1 (1 p(s 1, λ))(1 p(s 1, λ)) either one or both of its members will encounter a job opportunity in the market. The lifetime utility for a household with the first member employed solves 17

18 the following functional equation: V en (a, ɛ, λ, Γ) = max u(c t ) k(s 2 ) Φ(h) a a,s 2 [ + β (p(s2, λ)(1 χ(λ)) Q ee (a, ɛ, λ, Γ ) ɛ,λ + p(s 2, λ) χ(λ) Q ne (a, ɛ, λ, Γ ) + (1 p(s 2, λ)) (1 χ(λ)) Q en (a, ɛ, λ, Γ ) + (1 p(s 2, λ))χ(λ) V nn (a, ɛ, λ, Γ ) ] dπ ɛ ɛ dπ λ λ) (3.10) subject to a = R λ,γ a + w λ,γ hɛ 1 c. (3.11) Given the level of search intensity s 2 with a probability p(s 2, λ)(1 χ(λ)) the family will have both members employed next period. Moreover, with a probability p(s 2 )χ(λ) the family members will alternate roles in the labor market and the second agent will be allocated to market work (if he takes up on the offer). For the sake of brevity we omit the object V ne since the recursive representation is similar to that of equation (3.10). Finally, the lifetime utility for a household with both members employed solves the functional equation: V ee (a, ɛ, λ, Γ) = max u(c t ) Φ(h) (3.12) a a i [ + β (1 χ(λ)) 2 Q ee (a, ɛ, λ, Γ ) + χ(λ) 2 Q nn (a, ɛ λ, Γ ) ɛ,λ ] + (1 χ(λ)) χ(λ)(q en (a, ɛ, λ, Γ ) + Q ne (a, ɛ, λ, Γ )) dπ ɛ ɛdπ λ λ a = R λ,γ a + w λ,γ h i ɛ i c (3.13) Competitive Equilibrium The definition is similar to that of section 3.1 and for the sake of brevity we relegate it to the appendix. 3.3 Couples economy: Husbands and Wives The model of this section introduces further heterogeneity. We populate our economy with males and females of equal measure and we assume that each household consists of a husband and a wife. As in the model of the previous section we assume that there is an aggregate (household) period utility function for consumption of the form u(c) but here we assume that household members can differ in terms of their search technologies, separation probabilities and the 18

19 value of leisure. In particular, we distinguish between p m (s) and p f (s), k m (s) and k f (s), Φ m (h) and Φ f (h) and χ m (λ) and χ f (λ). Subscripts m and f denote male and female spouses respectively. We also introduce differences in the earnings potential of household members; first we allow the idiosyncratic labor income processes to differ between males and females. We denote the productive endowments of the male and female spouse by ɛ m and by ɛ f respectively. Second, we assume that there are differences in the relative prices of male and female labor inputs, i.e. that there is a gender wage premium. Female wages are a fraction µ f < 1 of male wages for any given level of idiosyncratic productivity. These features are important for two reasons. First, in the data, see section 2, the labor force attachment of males and females is significantly different. 70% of the male population and 50% of the female population are employed and hence with these additions we can go a lot further in matching patterns of intra-household specialization that we see in the data. Second, by introducing differences in potential labor income between spouses we hope to get a more realistic account for the role of family self insurance than in the previous model with ex ante identical agents. The value functions for the model of this section are similar to equations (3.8), (3.10) and (3.12) in the previous section and for the sake of brevity we omit them (see section A.1 of the Appendix for details). 3.4 Discussion Our model builds on Chang & Kim (2007) and Gomes, Greenwood, & Rebelo (2001) who assess the labor market implications of models with heterogeneous agents and aggregate uncertainty. There, as well as here, the distribution of match (job) rents is governed by the idiosyncratic productivity endowments. And, according to their realizations agents adjust their labor market status in each period. Our model goes beyond this by adding the following features: we introduce both own productivity shocks ɛ and match quality (separation) shocks χ(λ). And we assume that search in the labor market is subject to a technology that maps search effort s into arrival rates of job offers p(s, λ). Why do we need a rich structure of shocks? Without these shocks we would not be able to match the worker flows which we summarize in Tables 2 and 3. In particular, without the match specific shock (χ(λ) )our model would not be able to match the observed flows from employment to non-employment, and without the labor productivity shock ɛ we would not be able to target the flows between inactivity and unemployment. To see this consider an economy where only χ(λ) shocks matter. Search intensity would be increasing over time. In this environment because agents run down their stock of wealth during non-employment, and flows from unemployment to inactivity would be zero. If, however, the only shock in our economy was 19

20 an ɛ shock then existing matches would only dissolve when the agent s own productivity has fallen. In that case it would only be coincidental if our model matched both the observed EU and EI flows. Put differently χ(λ) shocks add a valuable degree of freedom to our calibration that allows us to force productive agents from employment to unemployment and let those agents whose ɛ has fallen over the life on the job, match the observed employment to out of the labor force pattern. 11 The search technology. We adopt a parsimonious representation of the search technology. In particular we assume that there two levels of search intensity that a worker can exert s {s I, s U } where the subscripts I and U stand for inactive (out of labor force) and unemployment (active searchers) respectively. Associated with these choices are the following probabilities of receiving a job offer next period: p(s, λ) = { pi (λ) if s = s I p U (λ) if s = s U Further on the search costs are assumed to be of the form: k(s) = { 0 if s = si k if s = s U These discrete choices are enough to capture our division between workers that search actively, and hence are counted as unemployed, and those whose optimal choice of search does not translate into a large enough contact rate with potential employers and hence are considered out of the labor force workers. We give the following interpretation to our technology: p U (λ) and p I (λ) are treated as technological upper bounds to the number of matches that are possible each period from states U and I, respectively. When we increase the values of these parameters we also have to increase the variance of the ɛ shocks to keep the transition rates close to the data, since standard arguments imply that a mean preserving spread in the distribution of ɛ should make searchers more selective. Why do the probabilities change over the business cycle? The idea here is to have frictions in the background as in Mortensen & Pissarides (1994) without making their micro-foundations explicit. By changing p U (λ) and p I (λ) we replicate the behavior of aggregate labor market conditions, i.e. the availability of job opportunities, over the business cycle. Similarly, when we shift χ(λ) we try to match the cyclical patterns for job separations that we observe in the data. 11 With couples search intensity is a function of the productivity and employment status of the spouse. We show below that this is one of the reasons couples models perform much better in matching the worker flows. 20